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Discrete Mathematics and Applications最新文献

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Ergodicity of the probabilistic converter, a serial connection of two automata 概率转换器的遍历性，两个自动机的串行连接
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0034
I. A. Kruglov
{"title":"Ergodicity of the probabilistic converter, a serial connection of two automata","authors":"I. A. Kruglov","doi":"10.1515/dma-2021-0034","DOIUrl":"https://doi.org/10.1515/dma-2021-0034","url":null,"abstract":"Abstract The paper provides necessary and sufficient conditions for the the ergodicity of a serial connection of automata under which the output sequence of a substitution Mealy automaton is fed to the input of an output-free substitution automaton. It is shown that the condition of complete indecomposability of the state transition probability matrix of a Mealy automaton provides a sufficient condition for ergodicity of the probabilistic converter as a serial connection of automata. It is also shown that if the partial state transition functions of a Mealy automaton commute, then the condition of ergodicity of a serial connection is equivalent to that of both original probabilistic converters.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46221079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiaffine polynomials over a finite field 有限域上的多仿射多项式
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0038
S. Selezneva
{"title":"Multiaffine polynomials over a finite field","authors":"S. Selezneva","doi":"10.1515/dma-2021-0038","DOIUrl":"https://doi.org/10.1515/dma-2021-0038","url":null,"abstract":"Abstract We consider polynomials f(x1, …, xn) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f(x1, …, xn) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44799650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues 仿射类似物逼近q值逻辑函数对线性流形的限制
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0037
V. G. Ryabov
{"title":"Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues","authors":"V. G. Ryabov","doi":"10.1515/dma-2021-0037","DOIUrl":"https://doi.org/10.1515/dma-2021-0037","url":null,"abstract":"Abstract For a finite q-element field Fq, we established a relation between parameters characterizing the measure of affine approximation of a q-valued logic function and similar parameters for its restrictions to linear manifolds. For q > 2, an analogue of the Parseval identity with respect to these parameters is proved, which implies the meaningful upper estimates qn−1(q − 1) − qn/2−1 and qr−1(q − 1) − qr/2−1, for the nonlinearity of an n-place q-valued logic function and of its restrictions to manifolds of dimension r. Estimates characterizing the distribution of nonlinearity on manifolds of fixed dimension are obtained.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47237726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Large deviations of branching process in a random environment. II 随机环境中分支过程的大偏差。2
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-12-01 DOI: 10.1515/dma-2021-0039
A. V. Shklyaev
{"title":"Large deviations of branching process in a random environment. II","authors":"A. V. Shklyaev","doi":"10.1515/dma-2021-0039","DOIUrl":"https://doi.org/10.1515/dma-2021-0039","url":null,"abstract":"Abstract We consider the probabilities of large deviations for the branching process Zn in a random environment, which is formed by independent identically distributed variables. It is assumed that the associated random walk Sn = ξ1 + … + ξn has a finite mean μ and satisfies the Cramér condition E ehξi < ∞, 0 < h < h+. Under additional moment constraints on Z1, the exact asymptotic of the probabilities P (ln Zn ∈ [x, x + Δn)) is found for the values x/n varying in the range depending on the type of process, and for all sequences Δn that tend to zero sufficiently slowly as n → ∞. A similar theorem is proved for a random process in a random environment with immigration.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45763053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Diagnostic tests under shifts with fixed filling tuple 具有固定填充元组的轮班下的诊断测试
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0027
Grigorii V. Antiufeev
{"title":"Diagnostic tests under shifts with fixed filling tuple","authors":"Grigorii V. Antiufeev","doi":"10.1515/dma-2021-0027","DOIUrl":"https://doi.org/10.1515/dma-2021-0027","url":null,"abstract":"Abstract We consider a fault source under which the fault functions are obtained from the original function f(x̃n) ∈ P2n $begin{array}{} displaystyle P_2^n end{array}$ by a left shift of values of the Boolean variables by at most n. For the vacant positions of the variables, the values are selected from a given filling tuple γ̃ = (γ1, γ2, …, γn) ∈ E2n $begin{array}{} displaystyle E^n_2 end{array}$, which also moves to the left by the number of positions corresponding to a specific fault function. The problem of diagnostic of faults of this kind is considered. We show that the Shannon function Lγ~shifts,diagn(n), $begin{array}{} displaystyle L_{tilde{gamma}}^{rm shifts, diagn}(n), end{array}$ which is equal to the smallest necessary test length for diagnostic of any n-place Boolean function with respect to a described fault source, satisfies the inequality n2≤Lγ~shifts,diagn(n)≤n. $begin{array}{} displaystyle leftlceil frac{n}{2} rightrceil leq L_{tilde{gamma}}^{rm shifts, diagn}(n) leq n. end{array}$","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42813652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the complexity of monotone circuits for threshold symmetric Boolean functions 关于阈值对称布尔函数单调电路的复杂性
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0031
I. Sergeev
{"title":"On the complexity of monotone circuits for threshold symmetric Boolean functions","authors":"I. Sergeev","doi":"10.1515/dma-2021-0031","DOIUrl":"https://doi.org/10.1515/dma-2021-0031","url":null,"abstract":"Abstract The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( n $begin{array}{} sqrt n end{array} $), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43093321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Boolean analogues of the Pascal triangle with maximal possible number of ones 具有最大可能数目的帕斯卡三角形的布尔类比
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0029
F. M. Malyshev
{"title":"Boolean analogues of the Pascal triangle with maximal possible number of ones","authors":"F. M. Malyshev","doi":"10.1515/dma-2021-0029","DOIUrl":"https://doi.org/10.1515/dma-2021-0029","url":null,"abstract":"Abstract The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by s(s+1)2 $begin{array}{} displaystyle frac{s(s+1)}{2} end{array}$ elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 $begin{array}{} displaystyle frac{s(s+1)}{3} end{array}$⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47623873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups 有限亏缺法与构造阿贝尔群的正态和几乎正态问题
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0030
A. V. Menyachikhin
{"title":"The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups","authors":"A. V. Menyachikhin","doi":"10.1515/dma-2021-0030","DOIUrl":"https://doi.org/10.1515/dma-2021-0030","url":null,"abstract":"Abstract The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽2m (m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46061616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the average-case complexity of Boolean functions under binomial distribution on their domains 二项分布布尔函数在其定义域上的平均情况复杂度
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0028
A. V. Chashkin
{"title":"On the average-case complexity of Boolean functions under binomial distribution on their domains","authors":"A. V. Chashkin","doi":"10.1515/dma-2021-0028","DOIUrl":"https://doi.org/10.1515/dma-2021-0028","url":null,"abstract":"Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42189872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Large deviations of branching process in a random environment 随机环境中分支过程的大偏差
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0025
A. V. Shklyaev
{"title":"Large deviations of branching process in a random environment","authors":"A. V. Shklyaev","doi":"10.1515/dma-2021-0025","DOIUrl":"https://doi.org/10.1515/dma-2021-0025","url":null,"abstract":"Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Yn+1=AnYn + Bn, where A1, A2, … are independent identically distributed random variables and Bn may depend on {(Ak,Bk),0⩽k<n} $ {(A_k,B_k),0leqslant k lt n} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44535506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
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